Question

How do you use the rational root theorem to find the roots of `x^3-x^2+2x-2`?

Answer 1

Answer: The root of this polynomial is `1`.

The Rational Roots Theorem says that:

• if `P(x)` is a polynomial with integer coefficients
• and `p/q` is a root of `P` (i.e. `P(p/q) = 0` ),

then `p` is a factor of the constant term of `P` and `q` is a factor of the leading coefficient of `P`.

In our case, `P(x) = x^3-x^2+2x-2`. So, the constant term is `-2` and the leading coefficient is `1`.

First, let's write down all the factors of the constant term:
`+-1, and +-2`. These will be the possible values of `p`.

Next, let's write down all the factors of the leading coefficient:
`+-1`. These will be the possible values of `q`.

Now, let's write down the possible values of `p/q`:
`+-1/1, +-2/1`, which can be simplified as `+-1 and +-2`.

It can be easily verified that the only `p/q` that is a rational root of P is `1`.

(We can check this result by factoring P(x) as `x^2(x-1) +2(x-1) = (x-1)(x^2+2)`, which admits the solution `1`)

A graphical illustration can be seen below, by plotting the corresponding function:

graph{x^3 - x^2 + 2x - 2 [-10, 10, -5, 5]}